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Properties

Label 4-700e2-1.1-c1e2-0-7
Degree $4$
Conductor $490000$
Sign $1$
Analytic cond. $31.2428$
Root an. cond. $2.36421$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s − 5·9-s + 2·13-s − 4·16-s + 14·17-s − 10·18-s + 4·26-s − 10·29-s − 8·32-s + 28·34-s − 10·36-s + 4·37-s + 4·41-s + 49-s + 4·52-s + 12·53-s − 20·58-s − 16·61-s − 8·64-s + 28·68-s + 12·73-s + 8·74-s + 16·81-s + 8·82-s + 14·97-s + 2·98-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 5/3·9-s + 0.554·13-s − 16-s + 3.39·17-s − 2.35·18-s + 0.784·26-s − 1.85·29-s − 1.41·32-s + 4.80·34-s − 5/3·36-s + 0.657·37-s + 0.624·41-s + 1/7·49-s + 0.554·52-s + 1.64·53-s − 2.62·58-s − 2.04·61-s − 64-s + 3.39·68-s + 1.40·73-s + 0.929·74-s + 16/9·81-s + 0.883·82-s + 1.42·97-s + 0.202·98-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(490000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.2428\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 490000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.589126195\)
\(L(\frac12)\) \(\approx\) \(3.589126195\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.3.a_f
11$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.11.a_n
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.13.ac_bb
17$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.17.ao_df
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.29.k_df
31$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.a_cg
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.47.a_dh
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.59.a_s
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.a_fa
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.79.a_fd
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.a_fu
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.97.ao_jj
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.569343753087953468815992872802, −7.943950985696702485922224990231, −7.41192519294123246216805168593, −7.38451703018051703490900915946, −6.20892833382707396477507471574, −5.93898740434763322465740513982, −5.86948996746859435559692437938, −5.14936161735373762545181523253, −5.04950450614587838523509099308, −4.01558432174714569962208636972, −3.57752582126466520607951640819, −3.25831654996759112584151679894, −2.76472422088648867371864611007, −1.95044579129989398832305438850, −0.808304732563204183922021296054, 0.808304732563204183922021296054, 1.95044579129989398832305438850, 2.76472422088648867371864611007, 3.25831654996759112584151679894, 3.57752582126466520607951640819, 4.01558432174714569962208636972, 5.04950450614587838523509099308, 5.14936161735373762545181523253, 5.86948996746859435559692437938, 5.93898740434763322465740513982, 6.20892833382707396477507471574, 7.38451703018051703490900915946, 7.41192519294123246216805168593, 7.943950985696702485922224990231, 8.569343753087953468815992872802

Graph of the $Z$-function along the critical line